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An Introduction to the Theory of Formal Languages and Automata JANUA LINGUARUM STUDIA MEMORIAE NICOLAI VAN WIJK DEDICATA edenda curat C. H. VAN SCHOONEVELD Indiana University Series Minor, 192/1 FORMAL GRAMMARS IN LINGUISTICS AND PSYCHOLINGUISTICS VOLUME I An Introduction to the Theory of Formal Languages and Automata by W. J. M. LEVELT 1974 MOUTON THE HAGUE • PARIS © Copyright 1974 in The Netherlands Mouton & Co. N. V., Publishers, The Hague No part of this book may be translated or reproduced in any form, by print, photoprint, microfilm, or any other means, without written permission from the publishers Translation: ANDREW BARNAS Printed in Belgium by N.I.C.I., Ghent PREFACE In the latter half of the 1950's, Noam Chomsky began to develop mathematical models for the description of natural languages. Two disciplines originated in his work and have grown to maturity. The first of these is the theory of formal grammars, a branch of mathematics which has proven to be of great interest to informa­ tion and computer sciences. The second is generative, or more specifically, transformational linguistics. Although these disciplines are independent and develop each according to its own aims and criteria, they remain closely interwoven. Without access to the theory of formal languages, for example, the contemporary study of the foundations of linguistics would be unthinkable. The collaboration of Chomsky and the psycholinguist, George Miller, around 1960 led to a considerable impact of transforma­ tional linguistics on the psychology of language. During a period of near feverish experimental activity, psycholinguists studied the various ways in which the new linguistic notions might be used in the development of models for language user and language acquisi­ tion. A good number of the original conceptions were naive and could not withstand critical test, but in spiteof this, transformational linguistics has greatly influenced modern psycholinguistics. The theory of formal languages, transformational linguistics, psycholinguistics, and their mutual relationships are the theme of this work. Volume I is an introduction to the theory of formal languages and automata; grammars are treated only as formal systems, and no application of the theory, linguistic or other, is made. Volume II in turn deals with applications of those mathe- VI PREFACE matical models to linguistic theory. Volume III treats applications of grammatical systems to models of language user and language learner, as well as the formal questions which have arisen as a result of such applications. The material is cumulative: Volume II supposes a general understanding of Volume I, and Volume III refers to the subjects dealt with in Volumes I and II. Volumes II and III have their own preface, so we can now turn to some introductory remarks with respect to the present volume. Volume I, independent of the two following volumes, should be seen as an introduction to the theory of formal languages and automata. A number of similar introductions are available at the moment, but I have nevertheless undertaken the present work for three reasons. First, most available texts, because they suppose an acquaintance with sophisticated mathematical theories and methods, are beyond the reach of many students of linguistics and psychology. More often than not, Chomsky's and Miller's contributions to the Handbook of Mathematical Psychology prove too difficult for early graduate teaching. The present introduction is kept at a rather elementary level; a general knowledge of college mathematics will be sufficient to follow the text, although familiarity with the elements of set theory and statistics will certainly be an advantage. Second, existing introductions treat a number of subjects which have little obvious relation to linguistics or psychology. The linguist or the psychologist is obliged to make his own selection from among a series of topics which he does not yet understand, and he might search in vain for a treatment of topics which are especially relevant to his field. Probabilistic grammars and gram­ matical inference, for example, are not treated in any of the existing introductions. Special attention has been paid to these topics in the present volume, but matters not directly relevant to linguistics or psychology have not been completely excluded, as a balanced presentation of the theory sets its own demands. The third reason for writing this introduction is to supply readers of the two following volumes with a concise survey of the main notions of formal language theory used there. The subject PREFACE VII index of this volume can be used to find definitions of technical terms: definitions are indicated by italicized page numbers. Without the help and cooperation of many, these three volumes could not have been realized. A first version was written during a sabbatical year at The Institute for Advanced Study in Princeton, New Jersey. I am deeply grateful to Professor Duncan Luce and to The Institute for the invitation which made my stay possible. Much in this work is due to the help and insights of Professor George Miller, former director of the Harvard Center for Cognitive Studies, where the new psychology of language originated under his guidance. Thanks to him I was granted a Research Fellowship at the Center in 1965, and by happy coincidence, he too was at the Institute for Advanced Study when I was composing the text. His attentive advice was most useful, especially in the writing of the third volume. Likewise, regular discussions with Dr. Philip Johnson- Laird helped to clarify many of the psychological issues. Conver­ sations with Professor Aravind Joshi on the subject matter of the first two volumes were also enormously stimulating and enjoyable; I profited almost daily from his erudition in the fields of both formal systems theory and mathematical linguistics. Finally, I wish to express my gratitude to all those who have contributed by critically reading the text in the original Dutch version: Professor L. Verbeek, Dr. H. Brandt Corstius, Mr. R. Brons, Dr. G. Kempen, Dr. A. van der Ven, Mr. E. Schils, Mr. L. Noordman, Dr. A. De Wachter-Schaerlaekens, and Professor A. Kraak. Their remarks not only prevented the printing of many disturbing errors, but also led to many enriching additions to the text. March 1973 W. J. M. Levelt Nijmegen TABLE OF CONTENTS Preface v 1. Grammars as Formal Systems 1 1.1. Grammars, Automata, and Inference 1 1.2. The Definition of "Grammar" 3 1.3. Examples 6 2. The Hierarchy of Grammars 9 2.1. Classes of Grammars 9 2.2. Regular Grammars 12 2.3. Context-free Grammars 16 2.3.1. The Chomsky Normal-Form 17 2.3.2. The Greibach Normal-Form 19 2.3.3. Self-embedding 21 2.3.4. Ambiguity 25 2.3.5. Linear Grammars 26 2.4. Context-sensitive Grammars 27 2.4.1. Context-sensitive productions 27 2.4.2. The Kuroda Normal-Form 31 3. Probabilistic Grammars 35 3.1. Definitions and Concepts 35 3.2. Classification 37 3.3. Regular Probabilistic Grammars 38 3.4. Context-free Probabilistic Grammars 44 3.4.1. Normal Forms 45 3.4.2. Consistency Conditions 50 X TABLE OF CONTENTS 4. Finite Automata 53 4.1. Definitions and Concepts 54 . 4.2. Nondeterministic Finite Automata 60 4.3. Finite Automata and Regular Grammars 63 4.4. Probabilistic Finite Automata 68 5. Push-Down Automata 75 5.1. Definitions and Concepts 76 5.2. Nondeterministic Push-down Automata and Context- free Languages 81 6. Linear-Bounded Automata 91 6.1. Definitions and Concepts 92 6.2. Linear-bounded Automata and Context-sensitive Languages 96 7. Turing Machines 101 7.1. Definitions and Concepts 102 7.2. A few Elementary Procedures 105 7.3. Turing Machines and Type-0 Languages 106 7.4. Mechanical Procedures, Recursive Enumerability, and Recursiveness 110 8. Grammatical Inference 115 8.1. Hypotheses, Observations, and Evaluation 115 8.2. The Classical Estimation of Parameters for Proba­ bilistic Grammars 118 8.3. The "Learnability" of Nonprobabilistic Languages . 121 8.4. Inference by means of Bayes' Theorem 124 Historical and Bibliographical Remarks 131 Bibliography 135 Author Index 139 Subject Index 140 1 GRAMMARS AS FORMAL SYSTEMS 1.1. GRAMMARS, AUTOMATA, AND INFERENCE The theory of formal languages originated in the study of natural languages. The description of a natural language is traditionally called a GRAMMAR; it should indicate how the sentences of a language are composed of elements, how elements form larger units, and how these units are related within the context of the sentence. The theory of formal languages proceeds from the need to provide a formal mathematical basis for such descriptions. Chomsky, the founder of the theory, envisaged more than a simple refinement of traditional linguistic description. He was primarily concerned with a more thorough examination of the basis of linguistic theory. This involves such questions as "what are the goals of linguistic theory?", "what conditions must a grammar fulfill in order to be adequate in view of these goals?", and "what is the general form of a linguistic theory?" Without a formal basis, these and similar questions cannot be handled with sufficient precision. Volume II of this book will deal with these issues; it will be shown that a formal language can serve as a mathematical model for a natural language, while a formal gram­ mar can act as a model for a linguistic theory. From a mathematical point of view, grammars are FORMAL SYSTEMS, like Turing machines, computer programs, prepositional logic, theories of inference, neural nets, and so forth. Formal systems characteristically transform a certain INPUT into a par­ ticular OUTPUT by means of completely explicit, mechanically applicable rules. Input and output are strings of symbols taken 2 GRAMMARS AS FORMAL SYSTEMS from a particular alphabet or VOCABULARY. For a formal grammar the input is an abstract START SYMBOL; the output is a string of "words" which constitutes a "sentence" of the formal "language".
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